DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

of Einstein–Cartan gravity is a direct generalization
of the Einstein–Hilbert action of General
Relativity:
SEC =− 1
16πG
d 4 x √ −gg µν˜Rµν
where ˜Rµν is the (nonsymmetric) Ricci tensor
with torsion
˜Rµν =∂λ ˜Ɣ λ µν −∂ν ˜Ɣ λ µλ +˜Ɣ λ µν ˜Ɣ τ λτ −˜Ɣ τ µλ ˜Ɣ λ τν
and ˜Ɣ λ µµ is a non-symmetric affine connection
with torsion, which satisfies the metricity condition
˜∇µgαβ = 0. The above action can be
rewritten as a sum of the Einstein–Hilbert action
and the torsion terms, but those terms are
not dynamical since the only one derivative of
the torsion tensor appears in the action in a surface
term. As a result, for pure Einstein–Cartan
gravity the equation for torsion is Tλ µν = 0, and
the theory is dynamically equivalent to General
Relativity. If the matter fields provide an external
current for torsion, the Einstein-Cartan gravity
describes contact interaction between those
currents. See metricity of covariant derivative,
torsion.
Einstein equations The set of differential
equations that connect the metric to the distribution
of matter in the spacetime. The features
of matter that enter the equations are the stressenergy
tensor Tµν, containing its mass-density,
momentum (i.e., mass multiplied by velocity)
per unit volume and internal stresses (pressure
in fluids and gases). The Einstein equations are
very complicated second-order partial differential
equations (10 in general, unless in special
cases some of them are fulfilled identically) in
which the unknown functions (10 components
of the metricgµν) depend, in general, on 4 variables
(3 space coordinates and the time):
Gµν ≡ Rµν − 1
2 gµνR
= 8πG
Tµν
c2 Here Gµν is the Einstein tensor, Rµν the Ricci
tensor and R its trace, and G is Newton’s constant.
The factor c−2 on the right assumes a
choice of dimensions forGµν of [length] −2 and
© 2001 by CRC Press LLC
Einstein tensor
for Tµν of [mass/length 3 ]. It has been common
since Einstein’s introduction of the cosmological
constant to add gµν to the left-hand
side. Positive produces a repulsive effect at
large distances. Modern theory holds that such
effects arise from some other (quantum) field
coupled to gravity, and thus arise through the
stress-energy tensor. A solution of these equations
is a model of the spacetime corresponding
to various astronomical situations, e.g., a single
star in an otherwise empty space, the whole universe
(see cosmological models), a black hole.
Unrealistic objects that are interesting for academic
reasons only are also considered (e.g., infinitely
long cylinders filled with a fluid). In full
generality, the Einstein equations are very difficult
to handle, but a large literature exists in
which their implications are discussed without
solving them. A very large number of solutions
has been found under simplifying assumptions,
most often about symmetries of the spacetime.
Progress is also being made in computational solutions
for general situations. Through the Einstein
equations, a given matter-distribution influences
the geometry of spacetime, and a given
metric determines the distribution of matter, its
stresses and motions. See constraint equations,
gauge, signature.
Einstein–Rosen bridge A construction by
A. Einstein and N. Rosen, based on the
Schwarzschild solution, wherein at one instant,
two copies of the Schwarzschild spacetime with
a black hole of mass M outside the horizon were
joined smoothly at the horizon. The resulting 3space
connects two distant universes (i.e., it has
two spatial infinities) each containing a gravitating
mass M, connected by a wormhole. Subsequent
study showed the wormhole was dynamic
and would collapse before any communication
was possible through it. However, recent work
shows that certain kinds of exotic matter can stabilize
wormholes against such collapse.
Einstein summation convention See summation
convention.
Einstein tensor The symmetric tensor
Gab = Rab − 1
2 gabR
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